9 THE PLANCHEREL-GODEMENT THEOREM 385 space H is a mapping x H-» U(x) of F into <&(H) such that U(e) = 1H , U(xy) = U(x)U(y) and £/(**)=</(*)*. Let E be a Hilbert space and x H* T(x) a mapping of F into «5f(E). In order that there should exist a Hilbert space H which is the Hilbert sum of E and another Hilbert space F, and a representation x h-+ U(x) of F on H such that T(x) = PU(x) \ E for all x e F, where P is the orthogonal projection of H on E, it is necessary and sufficient that T should satisfy the following three conditions : (1) rO?)=lE and r(x*) === T(*)* for all*eE. (2) If g : r -»• E is any mapping such that g(x) = 0 for all but a finite set of ele- ments x of F, then E (T(x*y) • (3) If g : F ~> E is any mapping such that g(x) = 0 for all but a finite set of elements x e F, then for each z £ F there exists a constant M2 > 0 such that £ (T(x*z*zy)-g(y)\g(x)-)^UI- £ (T(x*y) • g(y) \ g(x)). (x. jOeFxr <x, jOeFxr Moreover, if U and H satisfy these conditions and are such that the elements U(x) 'f(x e F and /e H) form a total set in H, then the representation U is deter- mined up to equivalence (equivalence of representations being defined as in (15.5)). (To show that these conditions are sufficient, let G be the subspace of Er consisting of mappings g : F -> E such that g(x) = 0 for all but a finite set of elements x e F, and consider the form on G x G (g,K)^V(g,K) = £ (T(x*y)-g(y)\h(x)) (x, y) e r x r which is a positive hermitian form. If N is the set of all g e G such that B(#, g} ~ 0, then B induces a nondegenerate positive hermitian form (g, h)\-+(g\ h) = B(#, A) on G/N which makes G/N a prehilbert space. Assume that G/N is a dense subspace of a Hilbert space H0 . Define an injection j : G/N ->• Er as follows : for each g e G, the image/(^) of g is the mapping By transporting the scalar product on G/N by means of y, we have a structure of pre- hilbert space on /(G/N) and hence, extending by continuity, an isomorphism of H0 onto a Hilbert space H. Then define U(x) by the condition that U(x) *j(g) is the mapping Let A be a separable commutative Banach algebra with involution, having a unit element e. Let P be the set of positive linear forms on A (or of traces on A), which is a subset of the dual A' of the Banach space A (15.6.11). If /i, /2 e P, we write/! <^/2 to mean that /2 — /i e P. (a) If /,/0 are traces on A such that/0 ^/, show that there exists a sequence (^n) of elements of A such that/0(;v) = lim/Otfx) for all x e A. (If g is the bitrace correspond- n-+oo ing to /, note that /0 may be written in the form /0 = u ° irg , where u is a continuous linear form on H^ .) (b) Let P! be the set of traces /e P such that ||/|| =/(e) g 1 ; then Pa is convex and weakly compact. Show that the set of extremal points of PI (Section 12.15, Problem 5) is equal to (P n X(A)) u {0}. (Use (a) to show that if a character is a trace it mustsuch that b2 =